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Significance Tests:

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Significance Tests: THE BASICS Could it happen by chance alone? Statistical Inference Confidence Intervals Use when you want to estimate a population parameter ... – PowerPoint PPT presentation

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Title: Significance Tests:


1
Significance Tests
  • THE BASICS
  • Could it happen by chance alone?

2
Statistical Inference
  • Confidence IntervalsUse when you want to
    estimate a population parameter
  • Significance TestsUse when you want to assess
    the evidence provided by data about some claim
    concerning a population
  • AN OUTCOME THAT WOULD RARELY HAPPEN BY CHANCE IF
    A CLAIM WERE TRUE IS GOOD EVIDENCE THAT THE CLAIM
    IS NOT TRUE

3
Overview of a Significance Test
  • A test of significance is intended to assess the
    evidence provided by data against a null
    hypothesis H0 in favor of an alternate hypothesis
    Ha.
  • The statement being tested in a test of
    significance is called the null hypothesis.
    Usually the null hypothesis is a statement of no
    effect or no difference.
  • A one-sided alternate hypothesis exists when we
    are interested only in deviations from the null
    hypothesis in one direction
  • H0 ?0
  • Ha ?gt0 (or ?lt0)
  • If the problem does not specify the direction of
    the difference, the alternate hypothesis is
    two-sided
  • H0 ?0
  • Ha ??0

4
HYPOTHESES
  • NOTE Hypotheses ALWAYS refer to a population
    parameter, not a sample statistic.
  • The alternative hypothesis should express the
    hopes or suspicions we have BEFORE we see the
    data. Dont cheat by looking at the data first.

5
CONDITIONS
  • These should look the same as in the last chapter
    (for confidence intervals)
  • SRS
  • Normality
  • For meanspopulation distribution is Normal or
    you have a large sample size (n30)
  • For proportions--np10 and n(1-p)10
  • Independence

6
CAUTION
  • Be sure to check that the conditions for running
    a significance test for the population mean are
    satisfied before you perform any calculations.

7
Test Statistic
  • A test statistic comes from sample data and is
    used to make decisions in a significance test
  • Compare sample statistic to hypothesized
    parameter
  • Values far from parameter give evidence against
    the null hypothesis (H0)
  • Standardize your sample statistic to obtain your
    TEST STATISTIC

8
P-values statistical significance
  • The probability (computed assuming H0 is true)
    that the test statistic would take a value as
    extreme or more extreme than that actually
    observed is called the P-value of the test. The
    smaller the P-value, the stronger the evidence
    against the null hypothesis provided by the data.
  • Significant in the statistical sense doesnt
    mean important. It means simply not likely to
    happen just by chance.
  • The significance level a is the decisive value of
    the P-value. It makes not likely more exact.
  • If the P-value is as small or smaller than a, we
    say that the data is statistically significant at
    level a.

9
INFERENCE TOOLBOX (p 705)
DO YOU REMEMBER WHAT THE STEPS ARE???
Steps for completing a SIGNIFICANCE TEST
  • 1PARAMETERIdentify the population of interest
    and the parameter you want to draw a conclusion
    about. STATE YOUR HYPOTHESES!
  • 2CONDITIONSChoose the appropriate inference
    procedure. VERIFY conditions (SRS, Normality,
    Independence) before using it.
  • 3CALCULATIONSIf the conditions are met, carry
    out the inference procedure.
  • 4INTERPRETATIONInterpret your results in the
    context of the problem. CONCLUSION, CONNECTION,
    CONTEXT(meaning that our conclusion about the
    parameter connects to our work in part 3 and
    includes appropriate context)

10
Step 1PARAMETER
  • Read through the problem and determine what we
    hope to show through our test.
  • Our null hypothesis is that no change has
    occurred or that no difference is evident.
  • Our alternative hypothesis can be either one or
    two sided.
  • Be certain to use appropriate symbols and also
    write them out in words.

11
Step 2CONDITIONS
  • Based on the given information, determine which
    test should be used. Name the procedure.
  • State the conditions.
  • Verify (through discussion) whether the
    conditions have been met. For any assumptions
    that seem unsafe to verify as met, explain why.
  • Remember, if data is given, graph it to help
    facilitate this discussion
  • For each procedure there are several things that
    we are assuming are true that allow these
    procedures to produce meaningful results.

12
Step 3CALCULATIONS
  • First write out the formula for the test
    statistic, report its value, mark the value on
    the curve.
  • Sketch the density curve as clearly as possible
    out to three standard deviations on each side.
  • Mark the null hypothesis and sample statistic
    clearly on the curve.
  • Calculate and report the P-value
  • Shade the appropriate region of the curve.
  • Report other values of importance (standard
    deviation, df, critical value, etc.)

13
Step 4INTERPRETATION
  • There are really two parts to this step
    decision conclusion. TWO UNIQUE SENTENCES.
  • Based on the P-value, make a decision. Will you
    reject H0 or fail to reject H0.
  • If there is a predetermined significance level,
    then make reference to this as part of your
    decision. If not, interpret the P-value
    appropriately.
  • Now that you have made a decision, state a
    conclusion IN THE CONTEXT of the problem.
  • This does not need to, and probably should not,
    have statistical terminology involved. DO NOT
    use the word prove in this statement.

14
Example 1
  • Your buddy (Jake) claims to be an A student
    (meaning he has a 90 average). You dont know
    all of his grades but based on what you have seen
    you think this claim is an overstatement. You
    took a simple random sample of his grades and
    recorded them. They are 92, 87, 86, 90, 80,
    91. You also know that all his grades in the
    class have a standard deviation of 3.5.

15
Step 1
  • We want to determine whether Jake is accurate in
    his measure of his course grade.
  • Our null hypothesis is that Jake has a course
    average of 90.
  • Our alternative hypothesis is that Jakes course
    average is below a 90.
  • H0 ? 90
  • Ha ? lt 90

16
Step 2
  • Since we know the population standard deviation
    we will be performing a z-test of significance.
  • We were told that our selection of grades was an
    SRS of Jakes scores.
  • The box plot shows moderate left skewness. Our
    sample is not large so we must assume that the
    population of all of Jakes grades are
    approximately normal in distribution in order for
    our sampling distribution to be approximately
    normal. Using the IQR(1.5) method for
    determining outliers we see that there are no
    outliers in this sample of grades.
  • Provided Jake has at least 60 overall grades, we
    are safe assuming independence and using the
    necessary formula for standard deviation.

17
Step 3
  • A curve should be drawn, labeled, and shaded.
  • You can use the formula to calculate your z test
    statistic for this problem
  • ? In this case z -1.6330
  • Mark this on your sketch.
  • Based on our calculations the P-value is 0.0512.
  • , s3.5, n6

18
Step 4
  • Since there is no predetermined level of
    significance if we are seeking to make a
    decision, this could be argued either way. If
    Jake were correct about being an A student, we
    would only get a sample of grades with an average
    this low in roughly 5.1 of all samples.
  • There is not overwhelming evidence against H0,
    however, this is enough to convince me that H0
    can be rejected.
  • Our evidence may not be strong enough to convince
    Jake that he is wrong. However, based on this
    evidence, I do not believe Jake is accurate about
    his average being a 90. It doesnt appear that
    Jake is the A student he claims to be.

19
WARNINGS
  • Tests of significance assess evidence against H0
  • If the evidence is strong, reject H0 in favor of
    Ha
  • Failure to find evidence against H0 means only
    that data are consistent with H0, not that we
    have clear evidence that H0 is true
  • If you are going to make a decision based on
    statistical significance, then the significance
    level a should be stated before the data are
    produced.
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